Determining the Optimal FRP Mesh–ECC Retrofit Scheme for Corroded RC Structures: A Novel Multi-Dimensional Assessment Framework

Abstract

Reinforcement corrosion significantly reduces the load-bearing capacity, ductility, and energy dissipation of reinforced concrete (RC) structures, thereby increasing their seismic failure risk. To enhance the seismic performance of in-service RC structures, this study employs an FRP mesh–engineered cementitious composite (ECC) retrofitting method and develops a multi-objective optimization decision-making framework. A finite element model incorporating reinforcing steel corrosion, concrete deterioration, and bond–slip effects is first established and validated against experimental results. Based on this model, a six-story RC frame is selected as a case study, and eight alternative FRP mesh–ECC retrofitting schemes are designed. Five core indicators are quantified, namely annual collapse probability, expected annual loss, capital expenditure, carbon emissions, and downtime. The results indicate that FRP mesh–ECC retrofitting can significantly improve the seismic performance of corroded RC structures. The overall uniform retrofitting scheme (SCS-2) achieves the most significant improvements in seismic safety and economic performance, but they are associated with highest capital expenditure and carbon emission. Story-differentiated schemes (SCS-3 to SCS-6) provide a trade-off between performance enhancement and cost–emission control. While partial component-focused schemes (SCS-7 and SCS-8) cut cost and carbon but do not lower seismic downtime. Furthermore, the improved fuzzy-TOPSIS method with interval weights and Monte Carlo simulation indicates that the balanced scheme SCS-1 delivers the most robust performance across five dimensions, with a best probability close to 90%. The results confirm the potential of FRP mesh–ECC retrofitting at both component and structural levels and provide a practical reference for selecting seismic retrofitting strategies for existing RC structures.

1. Introduction

Reinforced concrete (RC) structures are widely used worldwide because they provide favorable mechanical performance and cost efficiency. During service life, reinforcing steel is often exposed to chloride attack and carbonation, making it highly prone to corrosion [1]. This process reduces the cross-sectional area of reinforcing steel [2] and decreases its yield strength and ductility [3,4]. Cracking and spalling of cover concrete are also induced, which weakens the bond between reinforcing steel and concrete [5]. Consequently, the load-bearing capacity and ductility of RC structures deteriorate significantly [6]. Previous studies have shown that corrosion causes stiffness degradation, reduces hysteretic energy dissipation, and promotes brittle failure modes [7]. As a result, the risk of structural failure and collapse under seismic loading is significantly increased [8,9]. For RC structures with long service periods or located in coastal, hot-humid, or high-salinity environments, corrosion is particularly severe. It has become a critical factor restricting durability and seismic safety. Therefore, reducing corrosion-induced deterioration and restoring or improving seismic performance remain urgent challenges in structural engineering.

To mitigate such deterioration, multiple retrofitting techniques have been developed, including steel jackets [10], external steel cages [11], externally bonded FRP sheets or plates [12,13], and concrete jackets [14]. While these methods can partially restore strength and ductility, they also present limitations. Steel jacketing is prone to secondary corrosion, FRP sheets are less effective in improving ductility, and conventional concrete jackets fail to eliminate brittle failure patterns. Recently, hybrid systems combining FRP mesh with engineered cementitious composites (ECC) have attracted increasing attention. ECC offers high ductility and excellent crack control, whereas FRP mesh provides high strength, low weight, and corrosion resistance. Their integration enhances stiffness and load capacity, while significantly improving energy dissipation and durability. Early work by Li [15] demonstrated the potential of ECC as an inorganic binder for FRP, showing superior load capacity and crack control compared with conventional concrete beams. Subsequent studies investigated CFRP mesh–ECC composites and confirmed their ability to restrain cracks and enhance structural performance [16]. Parallel to this, patch repair techniques have gained prominence, involving removal of spalled concrete, cleaning of corrosion products, application of anti-corrosion coatings, and repair with advanced mortars [17,18]. Further experimental and analytical studies have validated the feasibility of FRP mesh–ECC retrofits. Jiang and Sui [19] showed that retrofitted concrete columns exhibited strain-softening behavior and maintained integrity at failure. Hu et al. [20] found that hybrid CFRP–ECC retrofitted beams effectively suppressed crack propagation, reduced interfacial stress concentrations, and achieved higher cracking, yielding, and ultimate loads compared with conventional retrofits. Zhou et al. [21] extended this research to corroded coastal bridge piers, confirming that FRP–ECC overlays effectively mitigated seismic performance degradation, with higher corrosion rates leading to more severe losses. Zeng et al. [22] proposed and validated a column-strengthening method combining section curvilinearization with UHP-ECC and FRP confinement, testing 11 large-scale RC columns. The technique markedly increased load capacity, while existing stress–strain models tended to underestimate ultimate strength, indicating a need for model updates. Bi [23] further demonstrated that corroded specimens typically failed in shear, whereas retrofitted specimens shifted to ductile flexural failure, with confinement and ductility markedly enhanced.

Existing studies have verified the effectiveness of FRP mesh–ECC retrofitting at the component level. Retrofitted beams and columns, whether in flexure or compression, consistently show superior load capacity, ductility, and energy dissipation. However, most research has focused on mechanical performance improvement of single components, while analyses of seismic performance enhancement at the structural level remain limited. In practical engineering, retrofitting scheme selection often involves multiple factors, including safety, economy, environmental sustainability, and construction feasibility. Scientific ranking and prioritization of alternatives are therefore essential. Several scholars have explored retrofitting scheme selection for existing structures and proposed frameworks based on performance indices, multi-objective optimization, or fuzzy decision-making. For example, Omidian and Khaji [24] studied a typical RC office structure. They compared steel jackets, CFRP, and GFRP with different thicknesses, and optimized them using a genetic algorithm. Results indicated that appropriate retrofitting strategies not only reduce seismic response sensitivity but also enhance ductility indices under strong earthquakes, while considering retrofit cost. Dai et al. [25] developed OpenSEES models for intact, corroded, and FRP-retrofitted RC frames. They evaluated seismic risk and economic benefit under varying corrosion levels and retrofit strategies. Results showed greater effectiveness in highly corroded structures within developed regions, while neglecting service life led to overestimation of retrofit benefits. Choi et al. [26] applied a multi-objective genetic algorithm to determine the number of FRP layers required for shear-critical columns in a three-story frame. Optimization objectives included the coefficient of variation (COV) of inter-story drift and retrofit cost proportional to FRP volume. A balance between seismic performance and economic efficiency was achieved. Razmara Shooli et al. [27] proposed a hybrid GA–PSO model for performance-based design optimization of 2D RC frames. The objective function minimized total steel and concrete cost. Findings showed that integrating nonlinear static and dynamic analyses within the hybrid model not only reduced computational cost but also decreased material consumption. These studies provide valuable references for decision-making in structural retrofitting. Nevertheless, since FRP mesh–ECC retrofitting is an emerging technology, systematic investigations at the structural level remain limited. Comprehensive analyses addressing both structural safety and multi-dimensional constraints are still lacking.

To address this knowledge gap, this study adopts an FRP mesh–ECC composite retrofitting method and develops a multi-objective decision-making framework. A finite element model was established that incorporated reinforcing steel corrosion, concrete deterioration, and bond–slip effects, and its accuracy was verified against experimental results. Based on this model, a six-story RC frame is used as a case study. Eight alternative FRP mesh–ECC retrofitting schemes are designed, and five core indicators are quantified, including annual collapse probability, expected annual loss, initial cost, carbon emissions, and annual downtime. After calculating these indicators, a fuzzy TOPSIS method combined with Monte Carlo simulation was employed for evaluation, and the optimal scheme was identified.

2. Decision-Making Optimization Framework

This study proposes a comprehensive optimization framework based on multi-dimensional performance indicators, enabling scientific ranking and selection among alternative retrofitting schemes, as shown in Figure 1. The framework follows the principles of multi-objectivity, quantifiability, and interpretability. Four dimensions are considered in the framework, namely safety, economy, environmental sustainability, and functional recovery, with five indicators selected as the basis for assessment.

➢

Annual collapse probability (λ_Collapse): quantifies structural failure risk under earthquakes and serves as the core measure of seismic safety.

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Expected annual loss (EAL): evaluates direct and indirect economic losses over the life cycle and represents economic performance.

➢

Capital expenditure (CapEx): reflects financial investment during implementation and indicates static economic cost.

➢

Carbon emissions (C_em): measure the carbon footprint of material production, transportation, and construction, highlighting environmental impact and alignment with low-carbon goals.

➢

Annual downtime (T_down): includes both construction and earthquake-induced interruptions, reflecting resilience and functional recovery capacity.

On this basis, a fuzzy TOPSIS method [28,29] is introduced to construct the optimization framework. Unlike conventional TOPSIS, the improved method combines interval weights with Monte Carlo simulation to better represent uncertainty and preference variability in decision-making. Specifically, indicator weights are defined as intervals, and random sampling within these intervals generates multiple weight vectors, which are then used to compute the relative closeness distribution for each retrofitting scheme.

3. Case Study Building and Retrofitting Schemes

3.1. Structural Information

According to the current Chinese design codes [30,31], a six-story, three-bay RC frame is designed as the case study structure. The building is assumed to be located in Shanghai, China, and serves as an office building. The first-story height is 3.9 m, while all other stories have a height of 3.3 m. In accordance with the seismic design code, the design seismic intensity is set to seven degrees (0.1 g), the site class is type II, and the seismic group is Group 1. The plan and elevation layouts of the structure are shown in Figure 2. For loading conditions, the floor dead load and live load are both taken as 2 kN/m²; the roof dead load is 4 kN/m², the roof live load is 0.5 kN/m², the basic snow load is 0.5 kN/m², and the basic wind load is 0.75 kN/m². The concrete design strength grade is C30, and the reinforcing steel is HRB400E. The thickness of the concrete cover is 30 mm. The column cross-section dimensions are 600 mm × 650 mm, and the beam cross-section dimensions are 300 mm × 600 mm. Reinforcement details of beams and columns are also shown in Figure 2, where the stirrup diameter is 8 mm with a spacing of 250 mm.

3.2. Finite Element Model and Validation

3.2.1. Finite Element Model

The finite element (FE) model of the case study structure was developed using the OpenSEES platform. Beam and column components were modeled with nonlinear force-based beam–column elements, where plastic deformations were concentrated at plastic hinge regions near the element ends. To balance accuracy and efficiency, each beam and column was divided into six elements, and the mesh sensitivity analysis verified the adequacy of this discretization. A fiber-section model was adopted, in which longitudinal reinforcement, cover concrete, and confined core concrete were discretized into fibers to accurately represent nonlinear behavior of each constituent material, as shown in Figure 3. The Concrete01 model was used for cover and core concrete to capture compressive softening and unloading characteristics, while reinforcing steel was modeled with the Hysteretic material to represent cyclic hysteresis under repeated loading. For FRP mesh–ECC retrofitted structures, the retrofit layer was modeled with the ECC01 material to simulate the high ductility of ECC. The FRP mesh within ECC was equivalently represented as enhanced confinement by modifying the concrete constitutive model. Moreover, to incorporate bond–slip effects between reinforcing steel and concrete, ZeroLengthSection elements were assigned at beam–column ends, and the Bond_SP01 material was introduced to steel fibers. This enabled a more realistic simulation of anchorage slip and bond deterioration of reinforcing steel under seismic loading.

3.2.2. Models for Deterioration of Materials

In order to realistically capture the deterioration mechanisms of corroded RC members and the subsequent improvements brought by FRP–ECC retrofitting, several well-established constitutive models were adopted. The model proposed by Du et al. [31] was used to describe the strength reduction of reinforcing steel under corrosion, while the model of Coronelli and Gambarova [32] accounted for the compressive strength loss of corroded cover concrete. For stirrup-confined core concrete subjected to corrosion, the modified Mander-based model developed by Yu et al. [33] was employed. To represent the enhanced confinement provided by FRP–ECC overlays, the constitutive relationship proposed by Li [34] was utilized. Finally, the bond–slip degradation between reinforcing steel and concrete was incorporated using the model of Lowes et al. [35]. Together, these models allow the FE framework to comprehensively represent strength degradation, bond deterioration, and the confinement effects of FRP–ECC in corroded and retrofitted RC structures.

(1) Model for corroded reinforcing steel

To account for material degradation caused by corrosion of reinforcing steel, the models proposed by Du et al. [32] are employed:

$$f_{\mathrm{us}}=f_{\mathrm{u}0}(1-0.014\eta )$$

(1)

$$f_{\mathrm{ys}}=f_{\mathrm{y}0}(1-0.005\eta )$$

(2)

where η denotes the corrosion ratio of reinforcing steel; f_u0 and f_us represent the ultimate tensile strength before and after corrosion, respectively; f_y0 and f_ys denote the yield strength before and after corrosion.

(2) Model for corrosion-damaged cover concrete

For the reduction in compressive strength of cover concrete due to steel corrosion, the model developed by Coronelli and Gambarova [33] is applied:

$$f_{\mathrm{c}}^{\prime }={\displaystyle \frac{f_{\mathrm{c}}}{1+K\epsilon /{\epsilon }_{\mathrm{c}0}}}$$

(3)

where $f_{\mathrm{c}}^{\prime }$ is the reduced compressive strength of concrete; K is a coefficient related to bar diameter and surface roughness, taken as 0.08 for medium-diameter reinforcing steel; ε_c0 is the strain at the compressive strength f_c; ε₁ is the average transverse tensile strain of cracked concrete perpendicular to the applied compression, which can be calculated as:

$${\epsilon }_1=(b_{\mathrm{f}}-b_0)/b_0=n_{\mathrm{bars}}{\omega }_{\mathrm{cr}}/b_0$$

(4)

where b₀ is the original section width; b is the reduced width considering corrosion-induced cracking; n_bars is the number of longitudinal reinforcing bars in the considered direction; and w_cr is the total crack width at a given corrosion level.

(3) Model for corroded stirrup confined core concrete

For corroded stirrup-confined concrete, the model proposed by Yu et al. [34] is adopted. This model modifies Mander’s model to describe the stress–strain response of stirrup-confined concrete under corrosion. The stress–strain curve is primarily governed by peak stress f_cc, peak strain ε_cc, and ultimate strain ε_ccu. For the peak stress f_cc, it can be calculated as:

$$f_{\mathrm{cc}}=f_{\mathrm{c}}\left(-5.19+6.20\sqrt{1+{\displaystyle \frac{2.25f_1}{f_{\mathrm{c}}}}}-2{\displaystyle \frac{f_1}{f_{\mathrm{c}}}}\right)$$

(5)

where f_l is the effective lateral confining pressure provided by corroded stirrups, it can be expressed as:

$$f_1={\displaystyle \frac{1}{2}}{k}_{\mathrm{e}}{\rho }_{\mathrm{svc}}{f}_{\mathrm{yvc}}$$

(6)

where k_e denotes the confinement effectiveness coefficient; ρ_svc and f_yvc are the volumetric stirrup ratio and yield strength of corroded stirrups, respectively, they can be calculated as

$${\rho }_{\mathrm{svc}}=(1-{\eta }_{\mathrm{sv}}){\rho }_{\mathrm{sv}}$$

(7)

$$f_{\mathrm{yvc}}=(1-{\alpha }_{\mathrm{yv}}{\eta }_{\mathrm{sv}})f_{\mathrm{yv}}$$

(8)

where α_yv is the reduction factor for corroded stirrup yield strength, taken as 0.005.

For square sections with square stirrups, the confinement effectiveness coefficient k_e is calculated as:

$$k_{\mathrm{e}}={\displaystyle \frac{1}{1-{\rho }_{\mathrm{sv}}}}\left(1-{\displaystyle \sum _{i=1}^n\frac{w_i^2}{6b_{\mathrm{c}}{h}_{\mathrm{c}}}}\right)\left(1-{\displaystyle \frac{s}{2b_{\mathrm{c}}}}\right)\left(1-{\displaystyle \frac{s}{2h_{\mathrm{c}}}}\right)$$

(9)

where ρ_sv is the longitudinal reinforcement ratio in the confinement direction; s is the stirrup spacing; w_i is the clear spacing between the ith pair of longitudinal bars; b_c and h_c are the stirrup spacings along two section directions.

For the peak strain ε_cc, and ultimate strain ε_ccu, they can be calculated as

$${\epsilon }_{\mathrm{cc}}=(1-1.915\eta ){\epsilon }_{\mathrm{c}}\left[1+5\left({\displaystyle \frac{f_{\mathrm{cc}}}{f_{\mathrm{c}}}}-1\right)\right]$$

(10)

$${\epsilon }_{\mathrm{ccu}}=0.004+(1-\eta ){\displaystyle \frac{1.4{\rho }_{\mathrm{svc}}{f}_{\mathrm{yvc}}{\epsilon }_{\mathrm{cc}}}{f_{\mathrm{cc}}}}$$

(11)

(4) Model for FRP mesh—ECC confined core concrete

According to the model proposed by Li [35], the peak stress $f_{\mathrm{cc}}^{\mathrm{FRP}\text{–}\mathrm{ECC}}$ and peak strain ${\epsilon }_{\mathrm{cc}}^{\mathrm{FRP}\text{–}\mathrm{ECC}}$ of FRP mesh–ECC confined concrete can be expressed as:

$$f_{\mathrm{cc}}^{\mathrm{FRP}\text{–}\mathrm{ECC}}=f_{\mathrm{c}}\left(1+3.08{\left({\displaystyle \frac{f_{\mathrm{le}}}{f_{\mathrm{c}}}}\right)}^{0.68}\right)$$

(12)

$${\epsilon }_{\mathrm{cc}}^{\mathrm{FRP}\text{–}\mathrm{ECC}}={\epsilon }_{\mathrm{c}}\left(1+0.92{\left({\displaystyle \frac{f_{\mathrm{le}}}{f_{\mathrm{c}}}}\right)}^{0.59}{\left({\displaystyle \frac{{\epsilon }_{\mathrm{le}}}{{\epsilon }_{\mathrm{c}}}}\right)}^{0.61}\right)$$

(13)

where f_le denotes the effective lateral confinement stress provided by the FRP mesh, it can be calculated as:

$$f_{\mathrm{le}}=k_{\mathrm{s}}{k}_{\mathsf{\theta }}{\displaystyle \frac{2n{t}_{\mathrm{f}}{b}_{\mathrm{f}}{E}_{\mathrm{f}}{\epsilon }_{\mathrm{f}}}{D(s_{\mathrm{f}}+b_{\mathrm{f}})}}$$

(14)

where n is the number of FRP mesh layers; E_f is the elastic modulus of a single fiber filament; ε_f is the ultimate strain of a fiber filament; t_f and b_f are the equivalent thickness and width of a fiber bundle; sf is the net spacing between adjacent fiber bundles; k_s is the section reduction coefficient, taken as 1.02 for rectangular sections; k_θ is the correction factor for wrapping angle.

(5) Model for deterioration of bond–slip

The bond–slip behavior between corroded reinforcing steel and concrete is modeled using the constitutive relation proposed by Lowes et al. [36]. This model assumes constant bond stresses before and after yielding, with post-yield bond stresses being lower.

Before yielding (i.e., σ_s < f_y), the stress–slip relation is expressed as:

$$S={\displaystyle \frac{1}{8}}{\displaystyle \frac{f_{\mathrm{s}}^2}{{\tau }_{\mathrm{E}}E}}{d}_{\mathrm{b}}$$

(15)

After yielding (i.e., σ_s ≥ f_y), the stress–slip relation becomes:

$$S={\displaystyle \frac{1}{8}}{\displaystyle \frac{f_{\mathrm{ys}}^2}{{\tau }_{\mathrm{E}}{E}_{\mathrm{s}}}}{d}_{\mathrm{b}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{f_{\mathrm{s}}-f_{\mathrm{ys}}}{{\tau }_{\mathsf{\gamma }}}}{\displaystyle \frac{f_{\mathrm{ys}}}{E_{\mathrm{s}}}}{d}_{\mathrm{b}}+{\displaystyle \frac{1}{8}}{\displaystyle \frac{{(f_{\mathrm{s}}-f_{\mathrm{ys}})}^2}{{\tau }_{\mathsf{\gamma }}{E}_{\mathrm{h}}}}{d}_{\mathrm{b}}$$

(16)

where S is the slip length of reinforcing steel; σ_s is the bar stress; E_s is the elastic modulus of steel; E_h is the hardening modulus; d is the bar diameter; τ_E and τ_γ are the bond stresses before and after yielding, respectively.

According to the CEB-FIP Model Code 2010 [37], the bond stress of reinforcing steel and concrete can also be expressed as:

$${\tau }_E=1.8\sqrt{f_{\mathrm{c}}^{\prime }}$$

(17)

3.2.3. Model Validation

To assess the reliability of the numerical model, quasi-static tests reported by Dai et al. [38] were first adopted as benchmarks. These included three RC columns, including UC-0.1 (uncorroded), C-E10-0.1 (10% corrosion), and C-E20-0.1 (20% corrosion), which were used to examine the effect of reinforcing steel corrosion on seismic performance. Further validation was performed against the experimental study by Cao et al. [39], in which uncorroded RC columns retrofitted with two layers of FRP mesh and an ECC jacket were tested (i.e., specimen CN2/R), thereby providing reference data on the baseline performance of FRP–ECC strengthening in the absence of corrosion. In addition, the numerical results were compared with the program reported by Bi [22], which investigated three CFRP mesh–ECC retrofitted corroded RC specimens, including CG2-0.1-10% (10% corrosion with two mesh layers), CG2-0.1-20% (20% corrosion with two layers), and CG4-0.1-20% (20% corrosion with four layers)—covering distinct corrosion levels and strengthening quantities.

Figure 4 compares the experimental hysteresis curves with the numerical simulations for all specimens. It is clear that the model successfully reproduces the corrosion-induced degradation in strength and ductility, while for retrofitted columns it captures the recovery of load and deformation capacities associated with additional FRP mesh layers at a fixed corrosion level. Quantitatively, peak-load predictions show close agreement, with a maximum absolute error of 13.46% and the majority of specimens within 10%. These findings confirm that the model not only captures the deterioration associated with corrosion but also reliably reflects the performance improvements achieved by FRP mesh–ECC retrofitting.

3.3. Retrofitting Schemes

Assuming an initial corrosion ratio of 20% for the case study structure, eight different FRP mesh–ECC retrofitting schemes are proposed. In all schemes, retrofitting is mainly applied at the plastic hinge regions of component ends, with FRP mesh embedded within the ECC cover. The number of FRP mesh layers and the ECC thickness are considered as the primary retrofitting variables. The cross-sectional dimensions of all retrofitted specimens remain unchanged before and after retrofitting.

These eight schemes can be broadly divided into three categories, including overall uniform retrofitting (Schemes 1 and 2), story-differentiated retrofitting (Schemes 3~6), and partial component-focused retrofitting (Schemes 7 and 8). For the overall uniform retrofitting schemes, ECC and FRP mesh are applied to both beam and column ends to achieve comprehensive enhancement. In the story-differentiated retrofitting schemes, greater strengthening measures are applied to the lower stories, where inter-story drifts are more severe during earthquakes, while upper stories adopt lighter or alternative measures [40]. In the partial component-focused retrofitting schemes, column failures are considered the primary cause of collapse during earthquakes. Therefore, ECC and FRP mesh are applied only to column ends, while beam ends are repaired by replacing the cover concrete with cement mortar. The schematic diagrams of the retrofitting schemes are shown in Figure 5. The eight retrofitting schemes are described in detail as follows.

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Scheme 1 (SCS-1): Beam and column ends are uniformly retrofitted with a 20 mm ECC layer and two layers of FRP mesh.

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Scheme 2 (SCS-2): Beam and column ends are uniformly retrofitted with a 40 mm ECC layer and four layers of FRP mesh.

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Scheme 3 (SCS-3): Beam and column ends of the lower three stories are retrofitted with 40 mm ECC and four FRP mesh layers, while the upper three stories are retrofitted with 20 mm ECC and two FRP mesh layers.

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Scheme 4 (SCS-4): Column ends are retrofitted with 40 mm ECC and four FRP mesh layers, while beam ends are retrofitted with 20 mm ECC and two FRP mesh layers.

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Scheme 5 (SCS-5): Beam and column ends of the lower three stories are retrofitted with 20 mm ECC and two FRP mesh layers, while the upper three stories are repaired by replacing the cover concrete with cement mortar.

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Scheme 6 (SCS-6): Beam and column ends of the lower three stories are retrofitted with 40 mm ECC and four FRP mesh layers, while the upper three stories are repaired by replacing the cover concrete with cement mortar.

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Scheme 7 (SCS-7): Column ends are retrofitted with 20 mm ECC and two FRP mesh layers, while beam ends are repaired by replacing the cover concrete with cement mortar.

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Scheme 8 (SCS-8): Column ends are retrofitted with 40 mm ECC and four FRP mesh layers, while beam ends are repaired by replacing the cover concrete with cement mortar.

In all retrofitting schemes, the retrofitting region of beam and column components is uniformly set to 700 mm at the component ends. This length is determined based on existing plastic hinge length models. For ECC material, according to existing code [41], the compressive strength is assumed as 45 MPa, the tensile strength as 3.0 MPa, and the ultimate tensile strain as 3%. The FRP mesh material is assumed to be carbon fiber with an orthogonal bidirectional woven configuration. Its mechanical properties include a tensile strength of 3800 MPa, an elastic modulus of 240 GPa, and a fracture elongation of 1.5% [42]. In terms of geometry, each carbon fiber bundle has a width of about 2 mm. The equivalent thickness of a single CFRP mesh layer is about 0.044 mm, corresponding to an areal density of 80 g/m². To facilitate comparative analysis, ten finite element models were developed considering intact, corroded, and retrofitted conditions. The intact structure is denoted as UC, the corroded structure as C-20, and the eight retrofitted structures as SCS-1 to SCS-8.

4. Models for Decision Indicators

4.1. Annual Collapse Probability λ_Collapse

The first indicator focuses on structural safety. The annual collapse probability λ_Collapse links seismic hazard with fragility functions to provide a probabilistic measure of failure risk over the service life. The annual collapse probability λ_Collapse of a structure is obtained by combining the seismic hazard curve with the fragility function corresponding to the collapse state. Specifically, the seismic hazard function H(IM) describes the annual exceedance probability of an intensity measure (IM). It can be expressed in exponential form [43]:

$$H(IM)=k_0·I{M}^{-k}$$

(18)

where k₀ and k are site-related parameters. These parameters can be calculated as [44]:

$$k={\displaystyle \frac{\ln ({\lambda }_D/{\lambda }_M)}{\ln (I{M}_M/I{M}_D)}}$$

(19)

$$\ln \left(k_0\right)=\ln \left[{\lambda }_D{\left(I{M}_D\right)}^k\right]={\displaystyle \frac{\ln \left(I{M}_D\right)·\ln \left({\lambda }_M\right)-\ln \left(I{M}_M\right)·\ln \left({\lambda }_D\right)}{\ln \left(I{M}_D/I{M}_M\right)}}$$

(20)

where λ_D and λ_M are the annual exceedance probabilities of the design basis earthquake (DBE) and the maximum considered earthquake (MCE), respectively. Based on recurrence periods, they are 0.0021 and 0.0004. IM_D and IM_M are the intensity measures for DBE and MCE defined in seismic design codes.

Fragility functions for each limit state are assumed to follow a lognormal distribution, representing the conditional probability of collapse given an intensity measure [45]:

$$P\left(Collapse|IM\right)=\mathsf{\Phi }\left({\displaystyle \frac{\ln \left(IM\right)-\ln \left(m_R\right)}{{\beta }_R}}\right)$$

(21)

where m_R is the median value of the fragility function; β_R is the logarithmic standard deviation; Φ(·) is the standard normal distribution function.

Combining the above, the annual exceedance probability of structural collapse is obtained through convolution integration, as

$${\lambda }_{Collapse}={\displaystyle {\int }_0^{\infty }P\left(Collapse|IM\right)}{\displaystyle \frac{dH\left(IM\right)}{dIM}}dIM$$

(22)

To obtain the fragility functions for collapse states, seismic fragility analysis was conducted on the ten structural models under different conditions. It is also noted that the calculations of expected annual loss (EAL) and annual downtime (T_down) depend on fragility functions corresponding to various damage states. Therefore, all the structural damage states must be reasonably classified and defined. In this study, structural performance is divided into four limit states (LS1–LS4), which define five damage levels. including intact, slight, moderate, severe, and collapse. The maximum inter-story drift ratio is chosen as the damage index. Thresholds are determined using characteristic points of the pushover curve. The yield point defines LS1 and is determined by the equal energy method. The peak point of the pushover curve defines LS2. When load capacity decreases to 90% of the peak, LS3 is defined. When load capacity decreases to 80% of the peak, LS4 is defined, which corresponds to collapse.

A total of 100 ground motion records are used for fragility analysis [46]. They are classified into four groups based on magnitude M_w and source-to-site distance R [47], including low-magnitude short-distance (LMSD, 5.8 < M_w < 6.5, 13 < R < 30 km), high-magnitude short-distance (HMSD, 6.5 < M_w < 7.0, 13 < R < 30 km), low-magnitude long-distance (LMLD, 5.8 < M_w < 6.5, 30 < R < 60 km), and high-magnitude long-distance (HMLD, 6.5 < M_w < 7.0, 30 < R < 60 km). The M_w–R distribution and response spectra of the records are shown in Figure 6.

In this study, the multi-stripe analysis (MSA) method [48] is employed to derive the fragility curves of the structure. The spectral acceleration S_a(T₁, 5%) is selected as the intensity measure. All ground motion records are uniformly scaled according to the spectral acceleration at the fundamental period of the structure. The initial intensity is set to S_a = 0.1 g, at which the structure remains elastic. The intensity is then increased incrementally by 0.1 g up to 2.0 g. At each intensity level, the number of records exceeding the thresholds of different limit states is counted. The fragility function for each limit state is then fitted using the binomial distribution and the maximum likelihood estimation method. The parameter estimation process is expressed as [49]:

$$\left({\widehat{m}}_R,{\widehat{\beta }}_R\right)=\arg \underset{m_R,{\beta }_R}{\max }{\displaystyle \sum _{j=1}^m\left[z_j\ln \mathsf{\Phi }\left(\frac{\ln \left(x_j/m_R\right)}{{\beta }_R}\right)+\left(n_j-z_j\right)\ln \left(1-\mathsf{\Phi }\left(\frac{\ln \left(x_j/m_R\right)}{{\beta }_R}\right)\right)\right]}$$

(23)

where n_j is the number of ground motions analyzed at intensity level IM = x, and z_j is the number of exceedances.

4.2. Expected Annual Loss EAL

Beyond safety, economic consequences are of primary concern. The expected annual loss (EAL) aggregates direct and indirect costs associated with different damage states, offering a lifecycle measure of economic risk. The expected annual economic loss of a structure is calculated from the direct and indirect economic losses associated with different damage states under seismic loading. It can be expressed as:

$$EAL={\displaystyle \sum _{i=1}^4{\lambda }_{LS,i}\left(L_{Di}+L_{Ii}\right)}$$

(24)

where λ_LS,i is the annual exceedance probability of structural damage reaching level i.

The probabilistic seismic risk function can be derived by combining fragility functions and seismic hazard functions [50]:

$${\lambda }_{LS,i}=H\left(m_{R,i}\right)\exp \left[{\displaystyle \frac{1}{2}}{k}^2{\beta }_{R,i}^2\right]$$

(25)

where m_R,i and β_R,i are the median value and logarithmic standard deviation of the fragility function for limit state i.

The direct economic loss is divided into three parts, including structural loss, decoration loss, and property loss [51]. Quantitative analysis is conducted using corresponding loss ratios:

$$L_D=R_M+R_D+R_P$$

(26)

where L_D is the total direct economic loss ratio, defined as the total direct economic loss divided by the replacement cost; R_M is the structural loss ratio, defined as the structural loss divided by the replacement cost; R_D is the decoration loss ratio; R_P is the property loss ratio.

For the structural loss ratio R_M, values are assigned according to Chinese codes [52,53] and HAZUS [54]. They are 0%, 10%, 40%, 70%, and 100% for intact, slight, moderate, severe, and collapse damage states, respectively.

The decoration loss ratio R_D is calculated as:

$$R_D={\gamma }_1{\gamma }_2\eta R_{D0}$$

(27)

where R_D0 is the decoration loss proportion for the five damage states (5%, 20%, 40%, 80%, 100%) [53]; η is the ratio of decoration cost to replacement cost, taken as 40% in this study; γ₁ is the regional economic adjustment factor, taken as 1.3; γ₂ is the functional adjustment factor, taken as 1.1.

The property loss ratio R_P is expressed as [55]:

$$R_P={\gamma }_3{R}_{P0}$$

(28)

where R_P0 is the property loss proportion for the five damage states (0%, 0%, 5%, 30%, 80%) [56]; γ₃ is the ratio of total property value to replacement cost, taken as 100% in this study.

The indirect economic loss model follows GB/T 27932-2011 Assessment methods of earthquake-caused indirect economic loss [57]. It is estimated using a proportional factor ξ related to the regional economy and direct economic losses. The indirect economic loss L_I can be calculated as

$$L_I=\xi L_D$$

(29)

where ξ is the proportional factor, taken as 1.6 in this study.

The replacement cost of the structure is calculated from the total floor area and current construction price. Assuming a construction cost of 3000 CNY/m², the replacement cost is approximately 2.8 million CNY. This value is used as the baseline for subsequent loss assessment and risk evaluation.

4.3. Capital Expenditure CapEx

In addition to lifecycle risk, the immediate financial and environmental burdens of retrofit implementation should be considered. The capital expenditure of a structural retrofitting scheme refers to the one-time cost incurred during implementation. It generally includes direct costs and indirect costs. According to the relevant valuation norms [58] and considering the current market prices of retrofitting materials and labor costs, direct costs consist of material costs and labor costs [59]. The material costs mainly include the consumption and unit prices of ECC, FRP mesh, cement mortar, and other retrofitting materials. Labor costs are estimated based on construction complexity and man-hour quotas. Indirect costs include construction management, scaffolding, overheads, and taxes. These are generally taken as about 30% of the direct costs according to engineering practice. The capital expenditure of each retrofitting scheme can be expressed as:

$$CapEx=C_{\mathrm{mat}}+C_{\mathrm{lab}}+C_{\mathrm{ind}}$$

(30)

where C_mat is the material cost, calculated from retrofitting scope and unit prices; C_lab is the labor cost, usually estimated as 40–70% of the material cost; and C_ind is the indirect cost, taken as 30% of the sum of material and labor costs.

4.4. Carbon Emissions C_em

The calculation of carbon emissions for the retrofitting schemes follows the Chinese standard [60]. This framework is consistent with EN 15804 [61], considering A1–A4 stages of the life cycle, i.e., raw material production, transportation to site, and construction. A checklist-based accounting method was adopted. The total carbon emissions of each scheme, C_em, are divided into emissions from material production C_sc and transportation C_ys, as

$$C_{\mathrm{em}}=C_{\mathrm{sc}}+C_{\mathrm{ys}}$$

(31)

For the production stage, carbon emissions are calculated as:

$$C_{\mathrm{sc}}={\displaystyle \sum _{k=1}^n{M}_k{F}_k}$$

(32)

where C_sc is the carbon emission from material production (kgCO₂e); M_k is the consumption of the kth material; F_k is the emission factor of the kth material (kgCO₂e per unit). For ECC, the factor is taken as 220 kgCO₂e/t. For FRP, the factor is taken as 10.09 kgCO₂e/kg [62].

For the transportation stage, carbon emissions are calculated as:

$$C_{\mathrm{ys}}={\displaystyle \sum _{k=1}^n{M}_k{D}_k{T}_k}$$

(33)

where C_ys is the carbon emission from transportation (kgCO₂e); M_k is the consumption of the kth material (Unit: ton); D_k is the average transportation distance of the kth material, assumed as 500 km in this study; T_k is the emission factor per transport mode assumed as 0.1 kgCO₂e/(ton·km).

To enhance the generality of the results, we benchmarked the adopted Chinese factors against international references. The IPCC [63] default process emission factor for cement/clinker is about 500 kgCO₂/ton, which is comparable in magnitude to the Chinese baseline value for cementitious materials (about 735 kgCO₂/ton). For freight transport, the DEFRA-2024 dataset [64] reports an average factor of about 0.38 kgCO₂e/(ton·km) for diesel heavy goods vehicles, higher than the 0.10 kgCO₂e/(ton·km) adopted here but still of the same order of magnitude.

4.5. Annual Downtime T_down

Functional recovery is captured through annual downtime (T_down), which accounts for both retrofit-induced construction time and expected seismic interruption. This indicator reflects the resilience and serviceability of retrofitted structures. In this study, the annualized downtime of each retrofitting scheme is quantified as the sum of construction downtime and post-earthquake downtime, as

$$T_{\mathrm{down}}={\displaystyle \frac{T_{\mathrm{retrofit}}}{n_{\mathrm{life}}}}+{\displaystyle \sum _{i=1}^4{\lambda }_{L{S}_i}\mathbb{E}\left[T|L{S}_i\right]}$$

(34)

where T_retrofit is the one-time downtime caused by retrofitting construction, estimated using the construction productivity method; n_life is the remaining service life of the retrofitted structure, taken as 30 years in this study; E[T∣LS_i] is the expected recovery time after reaching damage state, according to FEMA P-58 [65], values of 2, 14, 60, and 360 days are adopted for slight, moderate, severe, and collapse states, respectively.

The calculation of T_retrofit is based on the relevant specification [58]. It also considers construction productivity of retrofitting materials and processes (efficiency and curing time of columns and beams) as well as inter-story switching losses, as

$$T_{\mathrm{retrofit}}=\max \left({\displaystyle \frac{A_c}{p_c·c_c}},{\displaystyle \frac{A_b}{p_b·c_b}}\right)+N·t_{cure}+N·t_{switch}$$

(35)

where A_c and A_b are the construction areas of columns and beams, respectively; p_c and p_b are the daily construction capacities for column and beam retrofitting, taken as 10 m²/day per crew; c_c and c_b are the number of crews, set to 1 in this study; t_cure is the curing time, taken as 3 days per story; t_switch is the inter-story switching time, taken as 0.5 days per story; N is the total number of stories.

5. Results and Discussion

5.1. Annual Collapse Probability Analysis

Figure 7 illustrates the seismic fragility curves obtained from Equation (23) for the intact structure (UCS), the corroded structure (CS), and the eight retrofitting schemes (SCS-1 to SCS-8). It can be observed that under the same limit state, the exceedance probability of the corroded structure CS is consistently the highest, indicating a significant deterioration in seismic safety. After FRP mesh–ECC retrofitting, the exceedance probabilities of all eight schemes are markedly lower than those of CS across all limit states. In some cases, the retrofitted schemes even outperform the intact structure. This indicates that appropriate retrofitting not only restores the original seismic capacity but can also achieve further enhancement.

Based on the values of m_R and β_R for each limit state, the seismic risk probabilities are calculated as shown in

Table 1. Seismic risk of structures under different limit states.

Structures	Limit States
	LS1 (10−4)	LS2 (10−4)	LS3 (10−5)	LS4 (10−5)
UCS	3.82	1.61	3.17	2.61
CS	6.51	2.96	7.14	5.75
SCS-1	2.97	1.19	2.51	1.91
SCS-2	2.27	0.92	1.99	1.29
SCS-3	2.76	1.11	2.42	1.73
SCS-4	2.57	1.02	2.21	1.59
SCS-5	5.34	2.20	4.88	3.40
SCS-6	4.15	1.81	3.78	2.74
SCS-7	4.03	1.71	3.54	2.51
SCS-8	3.06	1.23	2.68	1.99

. Under the collapse state (LS4), the annual exceedance probability of CS reaches 5.75 × 10⁻⁵, which is much higher than that of UCS at 2.61 × 10⁻⁵. This confirms that reinforcement corrosion severely undermines seismic safety. After FRP mesh–ECC retrofitting, all eight schemes show significantly reduced exceedance probabilities compared with CS, with some schemes performing better than UCS. For example, under LS4, SCS-2 achieves the lowest annual exceedance probability of only 1.29 × 10⁻⁵, demonstrating the greatest improvement in safety. This is followed by SCS-4 (1.59 × 10⁻⁵) and SCS-3 (1.73 × 10⁻⁵), both of which show clear improvements over UCS. These results indicate that overall uniform retrofitting or schemes prioritizing column-end enhancement can effectively improve collapse resistance. The performances of SCS-1 and SCS-8 are similar, at 1.91 × 10⁻⁵ and 1.99 × 10⁻⁵, respectively, both better than UCS. In contrast, SCS-5 (3.40 × 10⁻⁵) and SCS-6 (2.74 × 10⁻⁵) show some improvement compared with CS, but their collapse probabilities remain higher than UCS, reflecting limited effectiveness. This is mainly because these schemes adopt only mortar replacement or low ECC–FRP configurations in the upper stories, which fail to provide sufficient improvements in ductility and load-bearing capacity.

5.2. Expected Annual Loss Analysis

Figure 8 presents the EAL of the different structures. A comparison shows that the intact structure (UCS) has an EAL of 0.218 × 10⁴ CNY/year, while the 20% corroded structure (CS) increases to 0.423 × 10⁴ CNY/year, nearly doubling. This highlights the amplifying effect of reinforcement corrosion on long-term economic risk. After retrofitting, the EAL values of all schemes are significantly lower than those of CS. Among them, SCS-2 performs best, reducing the EAL to 0.124 × 10⁴ CNY/year, which is about 40% lower than UCS. This demonstrates the high efficiency of comprehensive beam–column enhancement in controlling economic risk. SCS-3 and SCS-4 show similar performance (0.15 and 0.14 × 10⁴ CNY/year), both effectively mitigating long-term economic risk. In contrast, SCS-5 records an EAL as high as 0.303 × 10⁴ CNY/year, only slightly better than CS. This indicates that localized retrofitting-strengthening only the lower stories while replacing cover concrete with mortar in the upper stories has limited effectiveness in controlling economic risk. The effects of SCS-6 and SCS-7 (0.24 and 0.23 × 10⁴ CNY/year) are better than CS. These values remain higher than UCS, showing that the unfavorable economic impact of corrosion cannot be fully offset.

5.3. Capital Expenditure Analysis

Figure 9 illustrates the capital expenditure of the eight retrofitting schemes, including material, labor, and indirect costs. Among them, SCS-2 has the highest total cost, about 201,700 CNY. This consists of 91,300 CNY for materials, 63,900 CNY for labor, and 46,600 CNY for indirect costs. Due to the largest material consumption and the most complex construction process, this scheme is significantly more expensive than the others. The total costs of SCS-4 and SCS-3 are 156,600 CNY and 137,900 CNY, respectively, placing them at a medium-to-high cost level. SCS-5 has the lowest cost, only 43,400 CNY. This is mainly because mortar is used in the upper stories to replace cover concrete, which substantially reduces material and labor demand. The total costs of SCS-6, SCS-7, and SCS-8 range from 91,000 to 117,000 CNY, representing medium-cost schemes that highlight the economic advantage of partial retrofitting. Further analysis indicates that retrofitting costs are highly sensitive to ECC thickness and the number of FRP mesh layers. Increases in these parameters lead to nonlinear rises in material and labor costs, which are particularly evident in full beam–column retrofitting schemes. In contrast, partial or story-differentiated schemes (SCS-5 to SCS-8) achieve a better balance between cost and performance. Although their seismic performance improvements are less significant than those of overall uniform retrofitting, they provide higher economic efficiency and are more suitable in contexts with limited budgets or where cost-effectiveness is prioritized.

5.4. Carbon Emission Analysis

Figure 10 summarizes the carbon emission results of the eight retrofitting schemes. Overall, differences in material configuration and component coverage lead to significant variations in emission levels. SCS-2 produces the highest carbon emissions, approximately 3613 kg CO₂e. This is mainly due to the combined use of thick ECC and multiple FRP mesh layers, both of which are high-impact materials that substantially increase total emissions. In contrast, SCS-5 generates the lowest emissions, only 903 kg CO₂e, which is only about 25% of SCS-2.

In terms of contribution breakdown, ECC and FRP mesh are the dominant sources of carbon emissions. For example, in the overall uniform retrofitting schemes (SCS-1 and SCS-2), FRP mesh accounts for 30–35% of emissions, while ECC accounts for 45–50%. The transportation stage contributes relatively little, about 10%. This indicates that, compared with optimizing transportation distance, reducing material consumption or selecting low-carbon alternatives has a more substantial effect on overall emission reduction. In addition, it can be seen that although SCS-2 and SCS-4 provide significant improvements in structural safety, they also result in the highest emission levels. By contrast, SCS-3, SCS-6, and SCS-8 fall within the medium range of 1800–3000 kg CO₂e, representing a compromise between enhancing structural reliability and controlling environmental performance.

5.5. Downtime Analysis

Figure 11 compares the construction downtime and seismic downtime of the eight retrofitting schemes. Overall, construction downtime is mainly governed by the amount of retrofitting work, while seismic downtime reflects the structural resilience level. A clear trade could be observed, as schemes with longer construction periods generally show lower seismic downtime, whereas schemes with shorter construction times tend to maintain weaker functionality under earthquakes.

Among the eight schemes, SCS-2, SCS-4, and SCS-8 exhibit the longest construction downtimes (31.60, 30.00, and 29.44 h/year). This is because thick ECC layers and multiple FRP mesh layers are applied to both beam and column ends, significantly extending the construction cycle. However, these schemes achieve the lowest seismic downtimes (about 0.182–0.267 h/year), indicating that high-strength materials and thick configurations can markedly enhance structural resilience. In contrast, SCS-5 shows the shortest construction downtime, only 22.32 h/year. This is because mortar replacement was adopted in the upper ends without ECC or FRP reinforcement, substantially reducing material and labor input. Nevertheless, its seismic downtime reaches 0.464 h/year, the highest among all schemes, suggesting that oversimplified retrofitting of upper components significantly reduces overall resilience. SCS-7 also has a relatively short construction downtime (26.56 h/year) because only column ends were retrofitted while beam ends were replaced with mortar. Its seismic downtime remains relatively high (0.345 h/year), showing limited resilience improvement. In addition, SCS-1 and SCS-3 represent compromise strategies. SCS-1 (20 mm ECC + two FRP mesh layers) shows a moderate construction downtime of 27.12 h/year and a seismic downtime of 0.256 h/year, reflecting a balanced performance. SCS-3 takes slightly longer (29.36 h/year), but its seismic downtime remains low (0.267 h/year), making it overall superior to SCS-1. In comparison, SCS-6 has a construction downtime similar to SCS-1 (26.08 h/year), but due to insufficient retrofitting in the upper stories, its seismic downtime is significantly higher (0.372 h/year). This indicates that appropriate strengthening of upper components is crucial for minimizing lifecycle downtime losses.

5.6. Multi-Objective Scheme Optimization and Decision-Making

5.6.1. Multi-Objective Scheme Optimization

The optimization objective function in this study is formulated based on the five selected performance indicators. Since all indicators are defined as the smaller the better, the overall objective function can be expressed as

$$\min F=\left({\lambda }_{Collapse},EAL,CapEx,C_{em},T_{down}\right)$$

(36)

To prioritize alternative retrofitting schemes under multiple criteria, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is employed. The fundamental principle of TOPSIS is to construct a weighted normalized decision space, identify the positive ideal solution (where benefit-type indicators reach their maximum values and cost-type indicators their minimum values) and the negative ideal solution, and then evaluate the performance of each scheme according to its relative distance to these two reference points. The computational procedure is summarized as follows.

Assume that p schemes and q evaluation indicators are considered. The decision matrix can be expressed as:

$$X=\left(\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1q}\\ x_{21}& x_{22}& \dots & x_{2q}\\ ⋮& ⋮& \ddots & ⋮& \\ x_{p1}& x_{p2}& \dots & x_{pq}\end{array}\right)$$

(37)

where x_pq is the value of scheme p with respect to indicator q.

To remove dimensional inconsistency, the normalized decision matrix $R={\left[r_{pq}\right]}_{p\times q}$ can be obtained using Equation (38), as

$$r_{pq}={\displaystyle \frac{x_{pq}}{\sqrt{{\displaystyle {\sum }_{p=1}^m{x}_{pq}^2}}}}$$

(38)

Using the indicator weights w_q, the weighted normalized matrix $V={\left[v_{pq}\right]}_{p\times q}$ is then obtained as:

$$v_{pq}=w_q·r_{pq}$$

(39)

The positive ideal solution A⁺ and the negative ideal solution A⁻ are subsequently defined as:

$$A^{+}=\left\{\underset{p}{\max }{v}_{pq}|q\in Q_{\mathrm{benefit}};\underset{p}{\min }{v}_{pq}|q\in Q_{\mathrm{cost}}\right\}$$

(40)

$$A^{-}=\left\{\underset{p}{\min }{v}_{pq}|q\in Q_{\mathrm{benefit}};\underset{p}{\max }{v}_{pq}|q\in Q_{\mathrm{cost}}\right\}$$

(41)

where Q_benefit and Q_cost represent the sets of benefit-type and cost-type indicators, respectively.

The Euclidean distances of scheme p from the positive and negative ideal solutions, denoted as $S_p^{+}$ and $S_p^{-}$, are given by

$$S_p^{+}=\sqrt{{{\displaystyle \sum _{q=1}^n\left(v_{pq}-A_q^{+}\right)}}^2}$$

(42)

$$S_p^{-}=\sqrt{{{\displaystyle \sum _{q=1}^n\left(v_{pq}-A_q^{-}\right)}}^2}$$

(43)

The relative closeness coefficient C_p is then calculated as

$$C_p={\displaystyle \frac{S_p^{-}}{S_p^{+}+S_p^{-}}}$$

(44)

where a larger C_p value indicates that the scheme is closer to the positive ideal solution and farther from the negative ideal solution.

Schemes are therefore ranked in descending order of C_p, with the optimal scheme corresponding to the highest C_p. In conventional TOPSIS applications, the weight vector of evaluation indicators is typically predefined as fixed values. However, this approach presents several limitations when applied to the comprehensive evaluation of retrofitting schemes. First, the indicators often have heterogeneous dimensions (e.g., structural failure probability λ versus CapEx), making direct comparison prone to distortion. Second, weight assignment generally depends on expert judgment, which introduces subjectivity. Third, under uncertainty such as variability in decision-maker preferences or fluctuations in input parameters, fixed weights fail to adequately capture the robustness of scheme rankings.

To address these limitations, an improved fuzzy-TOPSIS method is proposed in this study, which integrates interval weights with Monte Carlo simulation. Specifically, the weights of the indicators are characterized as intervals. Monte Carlo sampling is then used to generate N sets of weight vectors within the defined intervals. For each weight vector, TOPSIS is performed to calculate the closeness coefficient C_p,k for each scheme. Based on these results, the best probability of scheme P_p is defined as:

$$P_p={\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N\mathbb{I}\left\{C_p^{\left(k\right)}=\max \left(C^{\left(k\right)}\right)\right\}}$$

(45)

where I{·} is the indicator function.

In this study, the interval weights of the five objective indicators are set as 0.30–0.50, 0.10–0.30, 0.10–0.30, 0.05–0.20, and 0.05–0.10, respectively. These intervals are based on a combination of literature review and informal surveys with practitioners and owners involved in retrofit projects. The rationale is that structural safety is always the foremost concern for owners, and thus receives the largest weight range. Economy follows, as both long-term lifecycle costs and upfront investment strongly influence the willingness to retrofit. Environmental impacts represented by carbon emissions are less frequently prioritized in developing countries. This is because the economic development often outweighs sustainability considerations. Therefore, a smaller interval is assigned. Lastly, functional recovery measured by annual downtime is given the lowest weight, since Section 5.5 shows that the differences among schemes are within the same order of magnitude and generally acceptable to owners.

After 10,000 Monte Carlo simulations, the best probability of each scheme is illustrated in Figure 12a. In addition, the distributions of closeness coefficients Cp and their probability densities for the eight retrofitting schemes are shown in Figure 12b and Figure 12c, respectively. It is clear that SCS-1 demonstrates the most robust performance among all schemes. Its mean closeness coefficient reaches 0.771 with a standard deviation of only 0.022, indicating that this scheme consistently remains close to the ideal solution under different weight combinations. Furthermore, its 5%, 50%, and 95% quantiles are 0.733, 0.773, and 0.805, respectively, reflecting a narrow distribution range and further confirming the stability of its ranking results. This robustness is closely related to its balanced performance across multiple dimensions, including safety (low λ_Collapse), economy (moderate EAL and CapEx), environmental sustainability (lower C_em), and functional recovery (reasonable control of T_down). Consequently, SCS-1 achieves the highest best probability of 89.3%, indicating that it is the preferred option in almost all scenarios.

SCS-7 exhibits some competitiveness in certain cases. Its mean closeness coefficient is 0.692, with a best probability of 6.9%. While its C_em and CapEx are relatively low, providing advantages in economy and environmental sustainability, its limited retrofitting only at column ends yields insufficient improvement in overall safety and resilience, resulting in weaker control of λ_Collapse and EAL. Accordingly, its distribution is wider, with a 5% quantile of 0.654 and a 95% quantile of 0.744, indicating lower robustness compared to SCS-1. It may only emerge as a favorable option when decision-making preferences strongly emphasize low cost and low carbon emissions. On the other hand, SCS-2, SCS-3, and SCS-4 achieve the best probability below 2%, with mean closeness coefficients ranging from 0.628 to 0.690. These schemes excel in enhancing safety, as reflected in the lowest λ_Collapse, but their use of thick ECC layers and multiple FRP meshes leads to higher CapEx and C_em, along with higher T_down. These drawbacks constrain their overall performance. Nevertheless, their 95% quantile values approach or exceed 0.78, indicating that they retain advantages when safety is given higher priority in the weighting scheme. By contrast, SCS-5, SCS-6, and SCS-8 perform poorly overall. SCS-5 exhibits the lowest mean closeness coefficient (0.372), reflecting significant shortcomings in safety and resilience. Although it demonstrates advantages in terms of low cost and reduced carbon emissions, these strengths are insufficient to compensate for its overall weaknesses. SCS-6 and SCS-8 achieve mean closeness coefficients of 0.608 and 0.657, respectively. However, their performance is limited due to unbalanced retrofitting strategies that emphasize only specific components or dimensions. As a result, their best probabilities remain zero.

The above findings show that schemes with a balanced configuration may perform consistently well across the five evaluation dimensions. This balance results in the highest level of robustness, with the best probability substantially higher than that of the other alternatives. In contrast, schemes with extreme configurations do not achieve competitive performance when multiple objectives are considered simultaneously. For example, SCS-2 involves high investment and high emissions, while SCS-5 minimizes cost at the expense of safety. These results indicate that a balanced trade-off among multiple objectives proves more effective than extreme optimization in a single dimension when selecting seismic retrofitting strategies.

5.6.2. Sensitivity Analysis Considering Interval Weight Uncertainty

To further examine the robustness of the decision-making framework, two additional sensitivity scenarios were introduced beyond the baseline intervals, namely the economy-priority case and the environment-priority case. In the economy-priority case, the weight ranges of the five indicators were set as 0.20–0.40, 0.20–0.40, 0.20–0.40, 0.05–0.15, and 0.05–0.10, respectively. In the environment-priority case, the weights were set as 0.20–0.40, 0.10–0.25, 0.05–0.20, 0.30–0.50, and 0.05–0.10. Figure 13 presents the best probabilities of each scheme under these two cases. The results indicate that the ranking changes significantly. In the economy-priority case, SCS-7 achieved the highest best probability (58.3%), followed by SCS-1 (41.7%). This reflects the advantage of SCS-7 in reducing upfront investment and carbon emissions, even though its improvement in safety is less significant than that of SCS-1. In the environment-priority case, the results show that SCS-7 overwhelmingly dominated with a best probability of 93.0%, while SCS-1 was selected as optimal in only 7.0% of the Monte Carlo simulations. This outcome is mainly attributable to SCS-7’s relatively low carbon emissions and moderate economic cost, which become decisive factors when decision preferences emphasize sustainability. In summary, the sensitivity analysis demonstrates that SCS-1 remains the most robust retrofit scheme under balanced or safety-priority weighting. However, when decision priorities strongly shift toward economy or environmental sustainability, SCS-7 can surpass SCS-1, emerging as the preferred option due to its cost-effectiveness and low-carbon advantage. These findings underscore the importance of stakeholder preferences and confirm that the proposed fuzzy-TOPSIS framework is capable of capturing such variations.

6. Conclusions

This study focuses on corroded RC frame structures and develops a multi-objective optimization decision-making framework based on the FRP mesh–ECC seismic retrofitting method. Five core indicators are quantified, including annual collapse probability (λ_Collapse), expected annual loss (EAL), capital expenditure (CapEx), carbon emissions (C_em), and downtime (T_down). On this basis, an improved fuzzy TOPSIS method integrating interval weights and Monte Carlo simulation is employed to perform comprehensive ranking and robustness analysis of eight FRP mesh–ECC retrofitting schemes. The main conclusions are as follows:

(1) For all eight FRP mesh–ECC retrofitting schemes, the annual exceedance probabilities under different limit states are lower than those of the corroded structure. Based on the annual collapse probability, comprehensive or column-prioritized retrofitting schemes markedly reduce collapse risk, and in some cases (e.g., SCS-2, SCS-3, SCS-4), their safety performance even surpasses that of the uncorroded structure. In contrast, schemes limited to strengthening only the lower stories provide limited performance improvement and may not restore the structure to its original condition.

(2) After retrofitting, the EAL values of all eight schemes decrease significantly. Among them, SCS-2 reduces the EAL by 70.7% compared with CS. However, schemes relying only on lower-story retrofitting (SCS-5 and SCS-6) or with insufficient column strengthening (SCS-7) show limited effectiveness in mitigating long-term economic risks.

(3) Although Scheme 2 achieves excellent performance in collapse resistance and EAL reduction, its CapEx reaches 201,700 RMB and its C_em are approximately 3613 kg CO₂e, both the highest among the eight schemes. From the component breakdown, ECC and FRP mesh are the major contributors to carbon emissions. In comprehensive retrofitting schemes (e.g., SCS-1 and SCS-2), FRP mesh accounts for 30–35% of total emissions, ECC for 45–50%, while transportation contributes only about 10%.

(4) In terms of annual downtime, the schemes demonstrate a clear trade-off between construction-related downtime and seismic-induced downtime. Comprehensive retrofitting schemes (e.g., SCS-2, SCS-4, and SCS-8) record the longest construction downtime (about 30 h/year) but achieve the lowest seismic downtime (0.182–0.267 h/year), reflecting higher resilience. In contrast, the lower-story scheme SCS-5 has the shortest construction downtime (22.32 h/year) but the highest seismic downtime (0.464 h/year), indicating clear deficiencies in functionality retention and rapid recovery.

(5) The results of the improved TOPSIS-based multi-objective optimization show that SCS-1 maintains a coordinated advantage across multiple dimensions, including safety, economy, environmental sustainability, and functional recovery. It achieves the highest mean closeness coefficient and the best probability close to 90%, demonstrating robustness far superior to the other schemes. Balanced performance across multiple objectives proves more effective than single-indicator optimization in seismic retrofitting strategies.

Unlike most existing studies which primarily examine FRP–ECC retrofitting at the component level, this research provides one of the first systematic structural-level evaluations. By integrating safety, economy, environmental sustainability, and functional recovery into a unified decision-making framework, the study bridges the gap between experimental validation and practical applications, thereby offering a novel reference for performance-based seismic retrofit of corroded RC structures. It should be noted that the FE model was primarily validated at the component level, and its application to the six-story RC frame represents a rational extension to structural-level assessment. Nevertheless, the scalability and reliability of the model for more complex structures and sites such as taller or irregular frames, long-span systems, regions with higher seismic hazard, and buildings with different occupancy/functional requirements remain to be further verified. Accordingly, the extrapolation of the present results beyond the studied configuration should be treated with caution. In addition, the detailed constructability aspects such as installation sequence, production rates, and waste factors were not explicitly modeled in this study, as they vary considerably with workforce, regional conditions, and seasonal factors. Instead, these practical influences can be incorporated into the framework through the CapEx and T_down indicators when applied to specific projects. Future work will incorporate broader calibration/validation (including subassembly or full-scale tests), alternative constitutive/aging models, and site-specific hazard–loss assumptions to enhance generality and robustness.